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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcvat | Structured version Visualization version GIF version |
Description: A nonzero subspace less than the sum of two atoms is an atom. (atcvati 30157 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatcvat.o | ⊢ 0 = (0g‘𝑊) |
lsatcvat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatcvat.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsatcvat.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcvat.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcvat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsatcvat.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lsatcvat.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
lsatcvat.n | ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
lsatcvat.l | ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
Ref | Expression |
---|---|
lsatcvat | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcvat.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
2 | lsatcvat.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lsatcvat.p | . . 3 ⊢ ⊕ = (LSSum‘𝑊) | |
4 | lsatcvat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
5 | lsatcvat.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec) |
7 | lsatcvat.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
9 | lsatcvat.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑄 ∈ 𝐴) |
11 | lsatcvat.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
12 | 11 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑅 ∈ 𝐴) |
13 | lsatcvat.n | . . . 4 ⊢ (𝜑 → 𝑈 ≠ { 0 }) | |
14 | 13 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
15 | lsatcvat.l | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) | |
16 | 15 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
17 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ¬ 𝑄 ⊆ 𝑈) | |
18 | 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17 | lsatcvatlem 36179 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
19 | 5 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑊 ∈ LVec) |
20 | 7 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
21 | 11 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑅 ∈ 𝐴) |
22 | 9 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑄 ∈ 𝐴) |
23 | 13 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
24 | lveclmod 19872 | . . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
25 | 5, 24 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) |
26 | lmodabl 19675 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
27 | 25, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Abel) |
28 | 2 | lsssssubg 19724 | . . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
29 | 25, 28 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
30 | 2, 4, 25, 9 | lsatlssel 36127 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
31 | 29, 30 | sseldd 3967 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
32 | 2, 4, 25, 11 | lsatlssel 36127 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
33 | 29, 32 | sseldd 3967 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
34 | 3 | lsmcom 18972 | . . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
35 | 27, 31, 33, 34 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
36 | 35 | psseq2d 4069 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) ↔ 𝑈 ⊊ (𝑅 ⊕ 𝑄))) |
37 | 15, 36 | mpbid 234 | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑅 ⊕ 𝑄)) |
38 | 37 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ⊊ (𝑅 ⊕ 𝑄)) |
39 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → ¬ 𝑅 ⊆ 𝑈) | |
40 | 1, 2, 3, 4, 19, 20, 21, 22, 23, 38, 39 | lsatcvatlem 36179 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
41 | 29, 7 | sseldd 3967 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
42 | 3 | lsmlub 18784 | . . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈)) |
43 | 31, 33, 41, 42 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈)) |
44 | ssnpss 4079 | . . . . . 6 ⊢ ((𝑄 ⊕ 𝑅) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅)) | |
45 | 43, 44 | syl6bi 255 | . . . . 5 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅))) |
46 | 45 | con2d 136 | . . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → ¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈))) |
47 | ianor 978 | . . . 4 ⊢ (¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)) | |
48 | 46, 47 | syl6ib 253 | . . 3 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))) |
49 | 15, 48 | mpd 15 | . 2 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)) |
50 | 18, 40, 49 | mpjaodan 955 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3935 ⊊ wpss 3936 {csn 4560 ‘cfv 6349 (class class class)co 7150 0gc0g 16707 SubGrpcsubg 18267 LSSumclsm 18753 Abelcabl 18901 LModclmod 19628 LSubSpclss 19697 LVecclvec 19868 LSAtomsclsa 36104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-oppg 18468 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-lcv 36149 |
This theorem is referenced by: lsatcvat2 36181 |
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